Polynomial Updates for the Unscented Kalman Filter

TL;DR

This paper introduces a Polynomial Unscented Kalman Filter that uses higher-order polynomial approximations and a Conjugate Unscented Transformation to improve state estimation accuracy in nonlinear spacecraft navigation scenarios.

Contribution

It proposes a novel polynomial approximation of the Bayesian update combined with CUT for better capturing higher-order moments in UKF.

Findings

Enhanced state estimation accuracy

Improved covariance consistency

Effective in non-Gaussian noise scenarios

Abstract

Most nonlinear filters used in spacecraft navigation are based on a linear approximation of the optimal minimum mean square error estimator. The Unscented Kalman Filter (UKF) handles nonlinear dynamics through a sigma-point transform, but the resulting state estimate remains a linear function of the measurement. This paper proposes a polynomial approximation of the optimal Bayesian update, leading to a Polynomial Unscented Kalman Filter that retains the structure of the standard UKF but enriches the measurement update with higher-order (polynomial) terms. To compute the moments required by this polynomial estimator, we employ a Conjugate Unscented Transformation (CUT), which accurately captures higher-order central moments of the state and measurement. Numerical examples, including Clohessy-Wiltshire and Circular Restricted 3-Body dynamics with non-Gaussian measurement noise, illustrate that the resulting polynomial-CUT filters improve both state estimation accuracy and covariance consistency when compared with their linear counterparts.